
\(\tau \rightarrow 3\mu \)
In the large \(\tan \beta \)  regime,
Higgs-mediated flavour violating diagrams
would be particularly important in this decay mode. The branching ratio can be expressed as [1]}, [2]}
\(\text{Br}(\tau \rightarrow 3\mu ) &=&\frac{G_F^2 \,m_\mu ^2 \,m_\tau ^7 \,\tau _\tau }{1536\,\pi ^3 \cos ^6\beta }\,|\kappa _{\tau \mu }^E |^2\left[\left(\frac{\sin (\alpha -\beta )\cos \alpha }{M_{H}^2} -\frac{\cos (\alpha -\beta )\sin \alpha }{M_{h}^2}\right)^2+\frac{\sin ^2\beta }{M_A^4}\right]\cr &\approx &\frac{G_F^2 \,m_\mu ^2\, m_\tau ^7\, \tau _\tau }{768\,\pi ^3\, M_A^4}|\kappa _{\tau \mu }^E |^2 \tan ^6\beta \,.\) 
In the above, \(\tau _\tau \)  is the \(\tau \)  life time and the approximate result has been obtained in the large \(\tan \beta \)  regime. For other Higgs-mediated lepton flavour violating 3-body decays,
\(\tau \rightarrow e\mu \mu \) ,
\(\tau \rightarrow 3e\)  or
\(\mu \rightarrow 3e\) , their corresponding branching ratios can easily be obtained with the appropriate
kinematic factors and the flavour changing factor \(\kappa \) .
While \(\text{Br}(\tau \rightarrow e\mu \mu )\)  can be as large as
\(\text{Br}(\tau \rightarrow 3\mu )\)  when \((Y^\dagger _\nu Y_\nu )_{13}\sim O(1)\) 
(which is possible in the case of an inverted hierarchical light neutrino spectrum),
other flavour
violating decays with final state electrons such as \(\mu \rightarrow 3e\) 
are
considerably suppressed due to the smallness of the Yukawa couplings.

\(B_s\rightarrow \ell _i \ell _j\)
\(B\)  mesons can also have Higgs-mediated LFV decays, which are significantly enhanced in the large \(\tan \beta \)  regime. The branching fraction is given by
\(\text{Br}(B_s\rightarrow \ell _i \ell _j) &=& \frac{G_F^4 \,M^4_W}{8\,\pi ^5}\,|V_{tb}^*V_{ts}|^2\, M_{B_s}^5 \,f_{B_s}^2\, \tau _{B_s} \biggl (\frac{m_b}{m_b+m_s}\biggr )^2 \nonumber \\[3mm] &\times &\sqrt{ \biggl [1-\frac{(m_{\ell _i}+m_{\ell _j})^2}{M_{B_s}^2}\biggr ]\biggl [1-\frac{(m_{\ell _i}-m_{\ell _j})^2}{M_{B_s}^2}\biggr ] }\nonumber \\[3mm] &\times &\Biggl \lbrace  \biggl (1-\frac{(m_{\ell _i}+m_{\ell _j})^2}{M_{B_s}^2}\biggr )|c^{ij}_{S}|^2+\biggl ( 1-\frac{(m_{\ell _i}-m_{\ell _j})^2}{M_{B_s}^2}\biggr ) |c_{P}^{ij}|^2\Biggr \rbrace  \,,\) 
where \(V_{ij}\)  represents the Cabibbo-Kobayashi-Maskawa (CKM) matrix,
\(M_{B_s}\)  and \(\tau _{B_s}\)  respectively denote the mass and lifetime of the
\(B_s\)  meson, while \(f_{B_s}=230\pm 30\)  MeV [3]} is the \({B_s}\)  meson
decay constant and \(c_{P}^{ij}\) , \(c_{S}^{ij}\)  are the form factors.
As an example, the lepton flavour violating (double-penguin) \(B_s\rightarrow \mu \tau \)  decay can be computed with the following form factors [2]}:
\(c_S^{\mu \tau }=c_P^{\mu \tau } &=&\frac{\sqrt{2}\,\pi ^2}{G_F \,M^2_W}\frac{m_\tau \, \kappa _{bs}^d \,\kappa _{\tau \mu }^{E \ast }}{\cos ^4\beta \,\bar{\lambda }^t_{bs}}\left[\frac{\sin ^2(\alpha -\beta )}{M^2_{H}}+\frac{\cos ^2(\alpha -\beta )}{M^2_{h}} + \frac{1}{M^2_{A}} \right]\nonumber \\[3mm]&\approx &\frac{8 \,\pi ^2 \,m_\tau \, m_t^2}{ M^2_W}\frac{ \epsilon _Y~ \kappa _{\tau \mu }^{E} ~\tan ^4\beta }{\left[1+(\epsilon _0+\epsilon _Y Y^2_{t} )\tan \beta \right]\left[1+\epsilon _0 \tan \beta \right]}\frac{1}{M^2_{A}} \,.\) 
Here, \(\kappa _{bs}^d\)  represents the flavour mixing in the quark
sector while \(\bar{\lambda }^t_{bs} = V^*_{tb}V_{ts}\) . Similarly,
\(\epsilon _0\)  and \(\epsilon _Y\)  are the down type quark form factors mediated
by gluino and squark exchange diagrams.
The final result was, once again, derived in the large \(\tan \beta \)  regime.
The branching fractions of other flavour violating decays such as
\(\text{Br}(B_{d,s}\rightarrow \tau e)\) , would receive identical contribution from the
Higgs penguins.
Likewise, the \(\text{Br}(B_{d,s}\rightarrow \mu e)\) 
can be calculated using the appropriate form factors and lepton masses;
as expected, these will be suppressed when compared to \(\text{Br}(B_{d,s}\rightarrow \tau \mu )\) .

\(\tau \rightarrow \mu P\)
Similar to what occurred in the previous processes, virtual Higgs exchange could also induce
decays such as \(\tau \rightarrow \mu P\) , where \(P\)  denotes a neutral pseudoscalar
meson (\(P=\pi , \eta , \eta ^{\prime })\) . In the large \(\tan \beta \)  limit, where the pseudoscalar Higgs couplings to down-type quarks are enhanced,
CP-odd Higgs boson exchanges provide the
dominant contribution to the \(\tau \rightarrow \mu P\)  decay. The coupling can be written as
\(-i (\sqrt{2} \,G_F)^{1/2}\tan \beta ~ A(\xi _d \,m_d \,\bar{d} \,d +\xi _s \, m_s \,\bar{s} \,s +\xi _b \,m_b \,\bar{b} \,b) + {\rm h.c.} .\) 
Here, the parameters \(\xi _d,\,\xi _s,\,\xi _b\)  are of order \(\mathcal {O}(1)\) .
Since we are mostly interested in the Higgs-mediated contributions, we estimate the amplitude of these processes in the limit when both \(\tau \rightarrow 3 \mu \)  and
\(\tau \rightarrow \mu P\)  are indeed dominated by the exchange of the scalar fields.
Accordingly, and following [5]}, one can write
\(\frac{\text{Br}(\tau \rightarrow \mu \eta )}{\text{Br}(\tau \rightarrow 3 \mu )}&\simeq &36 \,\pi ^2 \left(\frac{f^8_\eta \,m^2_\eta }{m_\mu \, m^2_\tau }\right)^2(1 - x_\eta )^2\left[\xi _s +\frac{\xi _b}{3}\left(1 +\sqrt{2} \,\frac{f^0_\eta }{f^8_\eta }\right)\right]^2 , \\\frac{\text{Br}(\tau \rightarrow \mu \eta ^{\prime })}{\text{Br}(\tau \rightarrow \mu \eta )}& \simeq & \frac{2}{9} \left( \frac{f^0_{\eta ^{\prime }}}{f^8_\eta }\right)^2\frac{m^4_{\eta ^{\prime }}}{m^4_\eta } \left(\frac{1 - x_{\eta ^{\prime }}}{1 - x_\eta } \right)^2\left[\frac{1 + \frac{3}{\sqrt{2}}\, \frac{f^8_{\eta ^{\prime }}}{f^0_{\eta ^{\prime }}}\left( \frac{\xi _s}{\xi _b} +\frac{1}{3}\right)}{\frac{\xi _s}{\xi _b} + \frac{1}{3} +\frac{\sqrt{2}}{3} \,\frac{f^0_\eta }{f^8_\eta }}\right]^2 , \\\frac{\text{Br}(\tau \rightarrow \mu \pi )}{\text{Br}(\tau \rightarrow \mu \eta )}& \simeq & \frac{4}{3} \left(\frac{f_\pi }{f^8_\eta }\right)^2 \,\frac{m^4_\pi }{m^4_\eta } ~(1 - x_\eta )^{-2}\left[\frac{\frac{\xi _d}{\xi _b} \,\frac{1}{1+z} + \frac{1}{2}\,(1 + \frac{\xi _s}{\xi _b})\frac{1-z}{1+z}}{\frac{\xi _s}{\xi _b} + \frac{1}{3} +\frac{\sqrt{2}}{3}\, \frac{f^0_\eta }{f^8_\eta }}\right]^2 \,,\) 
where \(z=m_u/m_d\) , \(m_\pi , \ f_\pi \)  are the pion mass and decay constant, \(m_{\eta ,\eta ^{\prime }}\)  are the masses of \(\eta ,\ \eta ^{\prime }\) , \(x_{\eta ,\eta ^{\prime }}= m_{\eta ,\eta ^{\prime }}^2/m_\pi ^2\) , and
\(f^8_{\eta ,\eta ^{\prime }}\)  and \(f^0_{\eta ,\eta ^{\prime }}\)  are evaluated from the corresponding matrix elements.
As first discussed in [6]}, and taking \(\xi _s,\xi _b \sim 1\)  and fixing the other parameters as in [5]}, one finds \(\frac{\text{Br}(\tau \rightarrow \mu \eta )}{\text{Br}(\tau \rightarrow 3 \mu )} \simeq 5\) . The other branching fractions such as
\(\text{Br}(\tau \rightarrow \mu \eta ^{\prime },\mu \pi )\)  are considerably suppressed compared to
\(\text{Br}(\tau \rightarrow \mu \eta )\) .
While the ratio \(\frac{\text{Br}(\tau \rightarrow \mu \eta ^{\prime })}{\text{Br}(\tau \rightarrow \mu \eta )}\) 
can be as large as \(6\times 10^{-3}\) ,
\(\frac{\text{Br}(\tau \rightarrow \mu \pi )}{\text{Br}(\tau \rightarrow \mu \eta )}\) 
would approximately lie in the range
\(10^{-3}-4\times 10^{-3}\)  [5]}.
Since all these ratios
are independent of \(\kappa _{\tau \mu }^{E}\) , the above quoted numbers can also be
applied to the present framework.
However, an enhancement in the
\(\text{Br}(\tau \rightarrow 3 \mu )\) , due to
the large values of \(\kappa _{\tau \mu }^{E}\) ,
would also imply sizeable values of \(\text{Br}(\tau \rightarrow \mu \eta )\) .

\(H_k\rightarrow \mu \tau \;(H_k = h,H,A)\)
The branching ratios of flavour violating
Higgs decays provide another interesting
probe of lepton flavour violation.
Following [9]}, the branching fraction \(H_k\rightarrow \mu \tau \) 
(normalised
to the flavour conserving one
\(H_k\rightarrow \tau \tau \) ) can be cast as:
\({\text{Br}(H_k\rightarrow \mu \tau )}= \tan ^2\beta ~ (|\kappa _{\tau \mu }^{E}|^2 )~ C_\Phi ~ { \text{Br}(H_k\rightarrow \tau \tau )} \, ,\) 
where we approximated \(1/\cos ^2\beta \simeq \tan ^2\beta \) .
The coefficients \(C_\Phi \)  are given by:
\(C_h = \left[\frac{\cos (\beta - \alpha )}{\sin \alpha }\right]^2, ~~~~C_H = \left[\frac{\sin (\beta - \alpha )}{\cos \alpha }\right]^2, ~~~~C_A = 1.\) 

